M1 MIDS/MFA/LOGOS |
| Année 2025 |
The purpose of this lab is to introduce linear regression using base R and the tidyverse. We work on a dataset provided by the MASS package. This dataset is investigated in the book by Venables and Ripley. This discusssion is worth being read. Our aim is to relate regression as a tool for data exploration with regression as a method in statistical inference. To perform regression, we will rely on the base R function lm() and on the eponymous S3 class lm. We will spend time understanding how the formula argument can be used to construct a design matrix from a dataframe representing a dataset.
# We will use the following packages.
# If needed, install them : pak::pkg_install().
stopifnot(
require("magrittr"),
require("lobstr"),
require("ggforce"),
require("patchwork"),
require("gt"),
require("glue"),
require("skimr"),
require("corrr"),
require("GGally"),
require("broom"),
require("tidyverse"),
require("ggfortify"),
require("autoplotly")
)Besides the tidyverse, we rely on skimr to perform univariate analysis, GGally::ggpairs to perform pairwise (bivariate) analysis. Package corrr provide graphical tools to explore correlation matrices. At some point, we will showcase the exposing pipe %$% and the classical pipe %>% of magrittr. We use gt to display handy tables, patchwork to compose graphical objects. glue provides a kind of formatted strings. Package broom proves very useful when milking lienar models produced by lm() (and many other objects produced by estimators, tests, …)
The dataset is available from package MASS. MASS can be downloaded from cran.
whiteside <- MASS::whiteside # no need to load the whole package
cur_dataset <- str_to_title(as.character(substitute(whiteside)))
# ?whitesideThe documentation of R tells us a little bit more about this data set.
Mr Derek Whiteside of the UK Building Research Station recorded the weekly gas consumption and average external temperature at his own house in south-east England for two heating seasons, one of 26 weeks before, and one of 30 weeks after cavity-wall insulation was installed. The object of the exercise was to assess the effect of the insulation on gas consumption.
This means that our sample is made of 56 observations. Each observation corresponds to a week during heating season. For each observation. We have the average external temperature Temp (in degrees Celsius) and the weekly gas consumption Gas. We also have Insul which tells us whether the observation has been recorded Before or After treatment.
Temperature is the explanatory variable or the covariate. The target/response is the weekly Gas Consumption. We aim to predict or to explain the variations of weekly gas consumption as a function average weekly temperature.
The question is wether the treatment (insulation) modifies the relation between gas consumption and external temperature, and if we conclude that the treatment modifies the relation, in which way?.
Have a glimpse at the data.
whiteside |>
glimpse()Rows: 56
Columns: 3
$ Insul <fct> Before, Before, Before, Before, Before, Before, Before, Before, …
$ Temp <dbl> -0.8, -0.7, 0.4, 2.5, 2.9, 3.2, 3.6, 3.9, 4.2, 4.3, 5.4, 6.0, 6.…
$ Gas <dbl> 7.2, 6.9, 6.4, 6.0, 5.8, 5.8, 5.6, 4.7, 5.8, 5.2, 4.9, 4.9, 4.3,…Even though the experimenter, Mr Whiteside, decided to apply a treatment to his house. This is not exactly what we call experimental data. Namely, the experimenter has no way to clamp the external temperature. With respect to the Temperature variable (the explanatory variable) we are facing observational data.
Before before proceeding to linear regressions of Gas with respect to Temp, perform univariate analysis on each variable.
Compare distributions of numeric variables with respect to categorical variable Insul
Gas and TempCompute the covariance matrix of Gas and Temp
Use ggpairs from GGally to get a quick overview of the pairwise interactions.
Build a scatterplot of the Whiteside dataset
Build boxplots of Temp and Gas versus Insul
Build violine plots of Temp and Gas versus Insul
Plot density estimates of Temp and Gas versus Insul.
Gas versus TempCompute slope and intercept using elementary computations
Overlay the scatterplot with the regression line.
lm()lm stands for Linear Models. Function lm has a number of arguments, including:
Use lm() to compute slope and intercept. Denote the object created by constructor lm() as lm0.
lm0 ?Including a rough summary in a report is not always a good idea. It is easy to extract tabular versions of the summary using functions tidy() and glance() from package broom.
For html output gt::gt() allows us to polish the final output
Function glance() extract information that can be helpful when performing model/variable selection.
R offers a function confint() that can be fed with objects of class lm. Explain the output of this function.
Plot a \(95\%\) confidence region for the parameters (assuming homoschedastic Gaussian noise).
Method plot.lm() of generic S3 function plot from base R offers six diagnostic plots. By default it displays four of them.
In order to obtain diagnostic plots as ggplot objects, use package ggfortify which defines an S3 method for class ‘lm’ for generic function autoplot (defined in package ggplot).
What are the diagnostic plots good for?
Load package ggfortify, and call autoplot() (from ggplot2) to build the diagnostic plots for lm0.
Generic function autoplot() called on on an object of class lm builds a collection of grapical objects that parallel the output of base R generic plot() function. The graphical objects output by autoplot() can be further tweaked as any ggplot object.
The diagnostic plots are built from the information gathered in the lm object returned by lm(...).
It is convenient to extract the required pieces of information using method augment.lm(). of generic function augment() from package broom.
Recall that in the output of augment()
.fitted: \(\widehat{Y} = H \times Y= X \times \widehat{\beta}\).resid: \(\widehat{\epsilon} = Y - \widehat{Y}\) residuals, \(\sim (\text{Id}_n - H) \times \epsilon\).hat: diagonal coefficients of Hat matrix \(H\).sigma: is meant to be the estimated standard deviation of components of \(\widehat{Y}\)Compute the share of explained variance
Plot residuals against fitted values
Fitted against square root of standardized residuals.
Hand-made normal qqplot for lm0
Gather the three hand-made diagnostic plots together using patchwork
Design a formula that allows us to take into account the possible impact of Insulation. Insulation may impact the relation between weekly Gas consumption and average external Temperature in two ways. Insulation may modify the Intercept, it may also modify the slope, that is the sensitivity of Gas consumption with respect to average external Temperature.
Have a look at formula documentation (?formula).
Check the design using function model.matrix(). How can you relate this augmented design and the one-hot encoding of variable Insul?
Display and comment the part of the summary of lm1 concerning estimated coefficients.
Comment the diagnostic plots built from the extended model using autoplot(). If possible, generate alternative diagnostic plots pipelining broom and ggplot2.
Function model.matrix() allows us again to inspect the design matrix.
In order to solve le Least-Square problems, we have to compute \[(X^T \times X)^{-1} \times X^T\] This can be done in several ways.
lm() uses QR factorization.
Extract the QR factorization of the design matrix from lm1
Q form an orthorgonal familyR is upper triangularCompute the Gram matrix of the row vectors of the design matrix for lm1
Understand .fitted column in augment(lm1, whiteside)
Try understanding the computation of .resid in an lm object. Compare .resid with the projection of Gas on the linear subspace orthogonal to the columns of the design matrix.
Understand .hat
Understanding .std.resid
residualsglance()Understanding column .sigma
This lab uses constructs of language R we have not mentioned so far.
Have a look at the documentation to get acquainted with this important and convenient notions.
S3 classes in RGeneric functions for S3 classes are used extensively in this lab. Functions from broom, ggplot2, and ggfortify are often generic functions.
methods(autoplot) lists the S3 classes for which an autoplot method is defined. Some methods are defined in ggplot2, others like autoplot.lm are defined in extension packages like ggfortify.
S4 classes in RThe output of autoplot.lm is an instance of S4 class.
See also New trends:
The S7 package is a new OOP system designed to be a successor to S3 and S4. It has been designed and implemented collaboratively by the R Consortium Object-Oriented Programming Working Group, which includes representatives from R-Core, Bioconductor, the tidyverse/Posit, and the wider R community.
tibbles with list columnsWe also used tibbles with list columns, that is nested dataframes. Nested dataframes show up in every implementation of dataframes (from R to Spark), see: Tidyr: article on nesting